If ∫(1/x+1/x^3)(√[23](3 x^-24+x^-26)) d x=-(α)/(3(α+1))(3 x^β+x^γ)^((α+1)/(α))+ C, x>0,(α, β, γ ∈ Z ), where C is the constant of integration, then…

If $\displaystyle \int\left(\frac{1}{x}+\frac{1}{x^3}\right)\left(\sqrt[23]{3 x^{-24}+x^{-26}}\right) \mathrm{d} x=-\frac{\alpha}{3(\alpha+1)}\left(3 x^\beta+x^\gamma\right)^{\frac{\alpha+1}{\alpha}}+\mathrm{C}, x>0,(\alpha, \beta, \gamma \in \mathbf{Z})$, where C is the constant of integration, then $\displaystyle \alpha+\beta+\gamma$ is equal to $\displaystyle \_\_\_\_$.